# Unique almost complex structure up to diffeomorphism

For which closed smooth manifolds does the action of the diffeomorphism group on the set of almost complex structures have exactly one orbit?

For example it is true for $$S^2$$.

You were lucky to find the only possible example. If you take any manifold of dimension $$\ge 4$$ you can pick an almost complex structure that is integrable in some closed ball and make it non-integrable outside of it. Since complex $$n$$-balls have more than one holomorphic structure, we are done. And all surfaces apart from $$S^2$$ have moduli of complex structures.
• Because you can choose an almost complex structure in a ball as you wish, and then extend to the whole manifold. Basically, when you have some almost complex structure on your manifold, you can take a point $p$ and a very small neighbourhood of it with some smooth coordinates. Then in this small neighbourhood $J$ is almost constant. You can perturb it to make it constant – Dmitri Panov Oct 7 at 12:28
On a $$2n$$-manifold $$M$$, the set of almost complex structures on $$M$$ are the sections of a smooth bundle $$\mathscr{J}(M)\to M$$ whose fibers are diffeomorphic to $$\mathrm{GL}(2n,\mathbb{R})/\mathrm{GL}(n,\mathbb{C})$$, a space of real dimension $$4n^2 - 2n^2 = 2n^2$$.
Thus, almost complex structures in dimension $$2n$$ depend locally on $$2n^2$$ functions of $$2n$$ variables, while diffeomorphisms of $$M$$ depend locally on $$2n$$ functions of $$2n$$ variables. Since $$2n^2>2n$$ when $$n>1$$, it follows that, when $$n>1$$, almost complex structures have local invariants, i.e., the diffeomorphism group cannot act transitively on the space of $$k$$-jets of almost complex structures for $$k$$ sufficiently large. Hence, not all almost complex structures can be equivalent under diffeomorphism when $$n>1$$, even locally.